Science has confirmed what vocalists already knew about how intervals affect singing in tune. Namely, singers find consonant intervals easier to tune than dissonant intervals.
In their barbershop quartet intonation study, Hagerman and Sundberg compared the standard deviations of each interval class. Then, they charted the relative tuning difficulty. They state, “The magnitude of the standard deviation for an interval is an indication of the difficulty to tune this interval… The smallest values appear as a function of the interval width.
“The smallest values appear for the fourth, the fifth, and the octave, i.e. for the ‘simplest’ intervals, if we by simplicity mean the number of partials, which, within a given frequency range, are common to the tones constituting the interval” (34).
Standard deviation formulas get used to measure statistical spread. In this study, the spread shows how closely the various interval widths within each interval class cluster around a center cent value. A smaller standard deviation shows that the singers found a particular interval easier to tune. Conversely, a bigger standard deviation indicates a harder to tune interval.